Tverberg-type theorems and the Fractional Helly property

Sätze vom Tverberg-Typ und die Fraktionale Helly-Eigenschaft

The main part of this thesis deals with Tverberg's theorem, the topological Tverberg theorem, and Sierksma's Dutch cheese conjecture. Using different approaches we obtain new Tverberg-type theorems: Lower bounds for the number of Tverberg partitions, and a Tverberg's theorem with constraints. The last chapter is devoted to the Fractional Helly property. There we obtain a topological Fractional Helly theorem. Possible generalizations of Kalai, Matoušek, and Meshulam towards families of bounded hoThe main part of this thesis deals with Tverberg's theorem, the topological Tverberg theorem, and Sierksma's Dutch cheese conjecture. Using different approaches we obtain new Tverberg-type theorems: Lower bounds for the number of Tverberg partitions, and a Tverberg's theorem with constraints. The last chapter is devoted to the Fractional Helly property. There we obtain a topological Fractional Helly theorem. Possible generalizations of Kalai, Matoušek, and Meshulam towards families of bounded homological VC-dimension are discussed. Helge Tverberg showed in 1966 that any set of (d+1)(q-1)+1 points in d-dimensional Euclidean space can be partitioned into q disjoint subsets such that their convex hulls have a non-empty intersection. Sierksma conjectured in 1979 that there is not only one such Tverberg partition, but at least ((q-1)!)^d many of them. Bárány, Shlosman, and Szücs generalized Tverberg's theorem for primes q towards the so-called topological Tverberg theorem for primes q. This result has been extended to prime powers q by several authors, e.g. Özyadin in 1986, and Volovikov in 1996. According to Matoušek, "the validity of the topological Tverberg theorem for arbitrary q is one of the most challenging problems in topological combinatorics". Chapter 1 comes with an extensive introduction to the subject, and to the tools needed in this thesis. In Chapter 2 we apply the equivariant method from topological combinatorics to obtain new Tverberg-type theorems. We show a lower bound for the number of Tverberg partitions for prime powers q combining the ansatz of Vucic and Zivaljevic with the method of Volovikov. Stimulated by the work of Schöneborn and Ziegler (2005), we introduce the concept of a constraint graph: Two adjacent vertices end up in different blocks of a Tverberg partition. This leads us to a generalization of the topological Tverberg theorem which we call "Tverberg's theorem with constraints". In our proof we obtain connectivity results of new chessboard-type complexes. A part from that, we extend the lower bound for the number of splittings of a generic necklace from Vucic and Zivaljevic to prime powers. In Chapter 3 we obtain for the first time a non-trivial lower bound for the number of Tverberg partitions that holds for arbitrary q. There we make use of a concept called Birch partitions which was introduced by Birch to prove Tverberg's theorem for d=2 in 1959. Birch and Tverberg partitions are closely related. We prove evenness and a lower bound for the number of Birch partitions. Parts of the result were stimulated by the results on the number of colorful simplices of Deza et al. (2005), and a computer project outlined in Chapter 4. Applying the topological Tverberg theorem with constraints, we show a lower bound for the number of Tverberg points for prime powers q. Combining this lower bound, and the lower bound for the number of Tverberg partitions from this chapter we improve once more the lower bound for the number of Tverberg partitions for prime powers q. This settles Sierksma's conjecture for a wide class of sets of points in the plane for q=3. Moreover, we discuss topological versions of the results on the number of Birch partitions. We come up with examples showing that these results do not immediately carry over to the topological setting. In Chapter 4 we discuss the outcome of a computer project which served us to look at many, many examples. This project motivated the results of Chapters 2 and 3, and it also led to a list of open problems. Chapter 5 is independent of the previous ones, and it deals with generalizations of the Fractional Helly theorem for finite families of convex sets due to Katchalski and Liu (1979). Our main result is a topological Fractional Helly theorem extending a result of Alon et al. (2003). The proof is based on a spectral sequence argument. On our way, we come up with a nice and short proof of the homological version of the nerve theorem due to Björner (2003). Moreover, we study the relation of the Fractional Helly property, and homological VC-dimension. This discussion is motivated by the Fractional Helly theorem for finite families of bounded VC-dimension due to Bárány and Matoušek (2003), and a technical report of Kalai (2004).…